Binomial branching process conditioned on extinction ($\tilde{p}=p\eta /(1-p+p\eta )$)

For offspring $\mathrm{Bin}(k,p)$ with $0<p<1$ and $kp>1$, let $\eta \in (0,1)$ be the branching extinction probability. Conditioning on extinction changes the offspring law to $\mathrm{Bin}(k,\tilde{p})$ with the displayed parameter. Indeed, the probability of $j$ children is multiplied by ${\eta }^{j-1}$, so the new probability generating function is $f(\eta s)/\eta$. Since $f(s)=(1-p+ps{)}^k$ and $f(\eta )=\eta$, this equals $((1-p+p\eta s)/(1-p+p\eta ){)}^k$. Its mean is $k\tilde{p}=f^{\prime }(\eta )<1$, because strict convexity of $f(s)-s$ and its zeros at $\eta$ and one force a negative derivative at the smaller zero. The conditioned process therefore has an exponentially decaying total-progeny tail. Unconditionally that tail instead converges to the survival mass $1-\eta$, with an exponentially small correction. When extinction has probability zero, conditioning on it does not define this law.